Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-128-33-c

·

A quartic del Pezzo fibration over $\mathbb P^2$.

  • $\pi_1\colon X\to Y$, onto a smooth complete intersection $Y$ of two quadrics in $\mathbb P^5$ (a Fano threefold of index $2$ and degree $4$), is a $\mathbb P^1$-bundle outside four points, over which the fibers are planes.

  • $\pi_2\colon X\to\mathbb P^2$ is a flat quartic del Pezzo fibration whose fibers are the hyperplane sections of $Y$ in a net; the discriminant is a reduced plane curve of degree $16$, over whose general point the fiber has one ordinary double point.

Defining data
Ambient
ℙ2P2 × ℙ5P5
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
2 copies of E^(2w3) + E^(w1 + w3)
Bundle
2 copies of O_2(2) + O_1(1) box O_2(1)
ZL62 label
2050.99w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 33
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 128
$\chi(X, \mathrm{T}_X)$ = -12

$\omega_X^\vee \cong \mathcal{O}_X(1)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 128
$\mathrm{c}_1^2\mathrm{c}_2$ = 128
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 4
Hodge diamond
1
00
0 2 0
0 2 2 0
0 0 6 0 0
0 2 2 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 12 0 x0 + 12 x2
$\mathrm{HH}^{ 3 }$ = x1 + 18 0 0 x1 18
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 9 0 0 x1 - x2 - 30 6 33
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 28t + 70t^{2} + 28t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(619)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(634)

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 4t + 1 \right) \left( 2t + 1 \right) ^{2} \left( 6t + 1 \right) ^{2} \left( 8048t^{4} + 384t^{3} - 280t^{2} + 32t - 1 \right) \left( 832941641498624t^{13} + 1041693078388736t^{12} + 635532318040064t^{11} + 370007298068480t^{10} + 169173610184960t^{9} + 46404969711616t^{8} + 7037952177408t^{7} + 577090813440t^{6} + 28570779360t^{5} + 1443608576t^{4} + 54062896t^{3} + 1079888t^{2} - 82359t + 136 \right)D^{6} + \left( 2t + 1 \right) \left( 6t + 1 \right) \left( 6757142445427173359616t^{20} + 13270200615783328907264t^{19} + 12172187771436452020224t^{18} + 8115144957826581397504t^{17} + 4570942560821719662592t^{16} + 1979508836877793640448t^{15} + 586722109341706731520t^{14} + 111083618714557599744t^{13} + 12487492571823067136t^{12} + 697257183110124544t^{11} + 1133808711252992t^{10} - 2715173938738944t^{9} - 221366831036672t^{8} + 6370937315008t^{7} + 2412087128384t^{6} + 134140839536t^{5} + 8348883568t^{4} + 142669468t^{3} + 5465620t^{2} - 245989t + 272 \right)D^{5} + \left( 675714244542717335961600t^{22} + 1641746682610976733265920t^{21} + 1864584885324864265650176t^{20} + 1453336021920531355795456t^{19} + 921842212829766020759552t^{18} + 474346137554291416694784t^{17} + 184221061160232867987456t^{16} + 51319381112409026691072t^{15} + 9993093385353509453824t^{14} + 1336161871747445178368t^{13} + 120586556529415766016t^{12} + 7218811442367514624t^{11} + 257236927540816896t^{10} - 4511413099374080t^{9} - 2069830984567296t^{8} - 218504885857152t^{7} - 12641495222720t^{6} - 375621704512t^{5} - 11293713280t^{4} - 329199592t^{3} - 4689312t^{2} + 149605t - 136 \right)D^{4} + t \left( 2837999827079412811038720t^{21} + 6447362578379848820981760t^{20} + 6931057376365728675921920t^{19} + 5298397427567553004175360t^{18} + 3362520192150231321935872t^{17} + 1704113434550347843567616t^{16} + 637523231228111165063168t^{15} + 169168194709654384345088t^{14} + 31430435510630456098816t^{13} + 4087182084386058190848t^{12} + 378250567487848071168t^{11} + 26217164310928740352t^{10} + 1444109635317749760t^{9} + 57745882109718016t^{8} + 889250626185216t^{7} - 65288141447936t^{6} - 5040434311680t^{5} - 15826338112t^{4} + 1391945920t^{3} - 163101936t^{2} + 1859924t + 425 \right)D^{3} + 8 t ^{2} \left( 783828523669552109715456t^{20} + 1686039076142934449455104t^{19} + 1734728914579828292714496t^{18} + 1310663897884981967454208t^{17} + 831646662125111248945152t^{16} + 413902919180624377872384t^{15} + 149280347520394687594496t^{14} + 37861196624900457811968t^{13} + 6732105161198102034432t^{12} + 849846681835444083712t^{11} + 79060927644603258880t^{10} + 5810217791263279360t^{9} + 350962264968651008t^{8} + 15575822570087808t^{7} + 336766097512768t^{6} - 9571723088848t^{5} - 841078609864t^{4} - 3594357380t^{3} + 357335256t^{2} - 8746433t + 129008 \right)D^{2} + 16 t ^{2} \left( 425699974061911921655808t^{20} + 878018749822211010330624t^{19} + 873612038578215315308544t^{18} + 655719343392377681739776t^{17} + 415699968791057400004608t^{16} + 203300263465968067526656t^{15} + 71015781405674106208256t^{14} + 17331961607124993773568t^{13} + 2969500708222087174144t^{12} + 365533283935732234752t^{11} + 34049326101576873984t^{10} + 2583240481675904768t^{9} + 161730793991616512t^{8} + 7349745310102912t^{7} + 167232348019904t^{6} - 3642004748656t^{5} - 307234517256t^{4} - 1520949822t^{3} + 125144760t^{2} - 142277t + 18904 \right)D + 16 t ^{2} \left( 173755091453841600675840t^{20} + 347986004366527372984320t^{19} + 338174525666949536415744t^{18} + 252938478874050579922944t^{17} + 160177484946929402773504t^{16} + 77234391138565585043456t^{15} + 26319527446732128583680t^{14} + 6236114884057849757696t^{13} + 1038277686735512403968t^{12} + 125353664745577606144t^{11} + 11675907404635080704t^{10} + 901301076036372224t^{9} + 57167365412426752t^{8} + 2605768815577216t^{7} + 60669443320320t^{6} - 1051112125728t^{5} - 86246590720t^{4} - 242741860t^{3} + 33241960t^{2} + 371059t + 2992 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ5P5

|λ| = 19.93774 λ ≈ 19.93774; multiplicity 1; Jordan blocks 1 λ ≈ 8 - 6.205616i; multiplicity 1; Jordan blocks 1 λ ≈ 8 + 6.205616i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 2; Jordan blocks 2 λ ≈ -4; multiplicity 1; Jordan blocks 1 λ ≈ -3.937741; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 2 2
Sector dimension
9
Spectral radius
$19.9377413$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$19.93774$11
$8 - 6.205616\mathrm{i}$11
$8 + 6.205616\mathrm{i}$11
$-6$22
$-4$11
$-3.937741$11
$-2$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 4\right) \left(\lambda + 6\right)^{2} \left(\lambda^{4} - 32 \lambda^{3} + 280 \lambda^{2} - 384 \lambda - 8048\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 4\right) \left(\lambda + 6\right)^{2} \left(\lambda^{4} - 32 \lambda^{3} + 280 \lambda^{2} - 384 \lambda - 8048\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 40 & 12 & 12 & 192 & 192 & 160 & 960 \\ 2 & 0 & 4 & 40 & 0 & 56 & 192 & 0 & 32 \\ 1 & 0 & 4 & 2 & 2 & 40 & 24 & 24 & 160 \\ 0 & 1 & 2 & 4 & 0 & 8 & 40 & 0 & 0 \\ 0 & 2 & 0 & 4 & 0 & 0 & 56 & 40 & 192 \\ 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 12 \\ 0 & 0 & 0 & 1 & 0 & 3 & 0 & 0 & 2 \\ 0 & 0 & 0 & 2 & 1 & -1 & 8 & 4 & 38 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 0\end{pmatrix}\notag\end{equation}$$